How to create a 'ribbon wrapped' sphere?
This shape being created in a 2D illustration program is not entirely possible in 3D without some distortion. That being said a close approximation may be achieved.



It's a bit of an involved process so I won't go into full details here, but I'll try to get the main steps. Here's the rough procedure of what I could come up with.


Create a bezier curve circle
Rotate it so it faces the front view
Rotate it 60º in the Z axis
Erase the back half
Duplicate it in edit mode so it is symmetrical along the Y axis
Repeat the process by duplicating the two half circles and rotating them 90º in Y axis so the produce the two frontal horizontal loops.
Now the tricky part is making sure you selecting the correct overlapping end vertex and connect them to the proper corresponding loose end by pressing the F key
Repeat the process 3 times. For the last closing loop just Toggle Cyclic option for the selected spline from the Properties Window
This will make half the necessary loops. Luckily the remaining ones are just a copy of the current ones rotated 45º along the Y axis, then scaled negatively -1 unit along the 7 axis to complete all the loops.




Afterwards you may add a flat segment bezier curve as Bevel Object to make the ribbon shape.

You may also optionally add a Shrinkwrap modifier to a sphere object to approximate the shape into a more accurate sphere.

If you want thickness a Solidify modifier after the Shrinkwrap will do the trick.